End-Effector Force Estimator

• End-effector force estimation is a technique that quantifies forces and wrenches at a robot's tool point using model-based, sensor-driven, and learning-based approaches.
• Analytical methods use physical modeling, kinematic constraints, and stiffness matrices to accurately infer reaction forces for safe and precise manipulation.
• Learning-based and sensor calibration pipelines integrate multimodal data, neural networks, and bias correction strategies to enhance real-time force prediction and adaptability.

An end-effector force estimator refers to any mechanism, algorithm, or integrated system capable of quantitatively predicting, measuring, or inferring the vector force (and possibly wrench) exerted at the active point of a robotic manipulator or device—whether during contact-rich interaction, teleoperation, compliant manipulation, or complex task environments. Estimators may use direct sensor measurements, model-based algorithms, or machine/deep learning approaches; their outputs are foundational for feedback control, safety, performance monitoring, and adaptive interaction in robotics, rehabilitation, manufacturing, and medical imaging.

1. Analytical and Model-Based Estimation Strategies

Physical modeling remains a core approach for end-effector force estimation, particularly in robot designs where the relationship between actuation inputs, mechanics, and output force is tractable. In "Design and integration of a parallel, soft robotic end-effector for extracorporeal ultrasound" (Lindenroth et al., 2019), the central paradigm is a kinetostatic equilibrium defined per soft fluidic actuator (SFA):

$w_\text{ext} = w_\theta + w_V$

where $w_\theta$ captures the elastic deformation forces/moments, and $w_V$ the reactions arising from constrained hydraulic volume. Both are mapped to the probe tip via configuration-dependent Jacobians ($J_\theta(x), J_V(x)$). The SFA is modeled as a Timoshenko beam, yielding a stiffness matrix ($K_\theta$) that is used—with local displacement—to estimate reaction forces/moments:

$\tau_\theta = K_\theta \delta x_\theta$

Further kinematic constraints (relating fluid volume change to tip displacement) result in a block-matrix system coupling tip displacement and internal reactions, from which the equilibrium solution is iteratively obtained.

For manipulators, similar physics-grounded models are frequently paired with forward/inverse kinematics or dynamic simulation (finite element methods in soft robotics (Cangan et al., 2022)). Orientation sensing, discrete actuation pressure, and comprehensive stiffness modeling (including fiber reinforcement) allow for the inverse estimation of external disturbance forces using quadratic programming; calibration of pressure scaling, stiffness coefficients and geometric nonlinearities is required for high-fidelity force estimates.

2. Learning-Based Force Estimation: Deep Neural and Multimodal Systems

Recent advances enable force estimation in scenarios where conventional direct measurements are impractical, unreliable, or cost-prohibitive. In robot-assisted surgery and human-robot interaction tasks, deep neural networks fuse vision and robot state for force prediction (Chua et al., 2020). Typical frameworks use ResNet50 for RGB image processing, concatenated with a feature-rich robot state vector (kinematics and intrinsic force/torque signals) and processed by fully-connected or convolutional layers. Multi-modal CNNs (as in (Hajian et al., 2022)) combine EMG and IMU time/frequency data for accurate biomechanical end-effector force estimation during dynamic contractions, achieving $R^2$ up to 0.91 intra-subject. Key benefits include generalization across tool, workspace, and material changes, and robust real-time inference ($>$30 Hz).

The learning-based paradigm extends to model-free estimation: in (Shan et al., 2023), joint currents, positions, velocities, and accelerations are input to MLP, LSTM, or CNN architectures. Carefully structured, diverse training data—including both free-space and rich contact tasks—allow for precise external wrench estimation, outperforming traditional model-based techniques. Transfer learning can adapt estimators to different manipulator models.

Table 1: Inputs and Estimation Techniques

Approach	Sensor/Signal Inputs	Estimation Strategy
Kinetostatic Model (Lindenroth et al., 2019)	SFA pressure, Jacobians, elastic response	Matrix inversion, stiffness theory
FEM + Sensors (Cangan et al., 2022)	Node positions, pressure, orientation sensors	Quadratic programming inverse dynamics
Deep Learning (Chua et al., 2020, Shan et al., 2023)	Vision, robot state, joint measurements	Convolutional/recurrent neural nets
EMG-IMU CNN (Hajian et al., 2022)	HD-EMG, multi-axis IMU	Multimodal feature-level deep fusion

3. Sensor-Driven, Calibration, and Bias Estimation Pipelines

Direct force/torque sensor readings (multi-axis F/T sensors) are common but present issues with bias drift, thermal effects, and mechanical stress. Automated continuous bias estimation (Nadeau et al., 2 Mar 2024) employs dual Kalman filters for online correction: the first predicts joint kinematic states, the second models bias and drift (assuming known end-effector inertial parameters). Kinematics are computed via exponential map and adjoint transformations, and task-space velocity/acceleration is mapped using the Jacobian. Bias is removed, yielding more reliable force signals for load identification, contact detection, and safe HRI. Such pipelines decouple sensor bias estimation from dynamic modeling and adapt over time.

4. Compliance, Nonlinearities, and Physical Interaction Safety

Physical compliance is essential for safe, repeatable force estimation and interaction. In ultrasound imaging, quasi-direct-drive actuators (QDD) (Chen et al., 4 Oct 2024) provide low backdrive torque, enabling both passive compliance and high-bandwidth ($100$ Hz) PID-based force regulation via current/torque mapping. Passive elements (e.g., timing belt transmissions) yield inherent impact absorption. Such systems exhibit low force tracking RMSE ($\sim0.83$ N) relative to standard articulated arms ($\sim4.7$ N), supporting stable interaction with dynamic tissues.

Soft robot end-effectors deliberately vary stiffness axially vs. laterally (e.g., high axial for probe stability, low lateral for safety during accidental movements (Lindenroth et al., 2019)). Features like braided nylon mesh mitigate twist, enhancing off-axis force predictability.

5. Application-Specific Estimation and Control Frameworks

End-effector force estimation is highly application-dependent. In distributed formation control (Wu et al., 2021), estimation is implicit via virtual springs (with potential energy gradients mapped to joint torques through the Jacobian) and disturbance compensation using embedded internal models. For agricultural end-effectors, calibration (e.g., strawberry peduncle gripping/cutting (S et al., 2022)) via experimental characterization yields safe force limits—grip under $10$ N to avoid damage, cut at $\sim15$ N with optimized blade wedge and orientation.

Adaptive arm support robots (Yang et al., 4 Apr 2024) estimate human joint angles by inverse kinematics from measured limb positions, transforming these to a reference frame and applying gravity loading models. The required support force is computed via Jacobian pseudo-inverse mapping and then projected to joint torques; experimental EMG measurements confirm reductions ($\sim57-60$%) in muscular effort.

In manufacturing, hybrid force-motion control benefits from real-time surface normal estimation through projection and friction compensation on F/T sensor data (Nasiri et al., 5 Apr 2024), using:

$$\hat{\mathbf{n}}_\text{surf} = \frac{\mathbf{f}_\text{s} - \mathbf{f}_\tau}{\|\mathbf{f}_\text{s} - \mathbf{f}_\tau\|}$$

where friction force $\mathbf{f}_\tau$ is derived from velocity and weighted average friction coefficient calculations.

6. Limitations, Calibration, and Generalization Issues

Force estimation accuracy is fundamentally limited by model fidelity, sensor calibration, and data diversity. Deep learning models manifest viewpoint/multimodal data sensitivity (Chua et al., 2020), while analytical and sensor-driven systems may suffer from unaccounted nonlinearities, variable material properties, and environmental noise. Calibration—whether for pressure-scaling factors in soft robots (Cangan et al., 2022) or interference rejection in multi-modal sensor arrays (Tanaka et al., 23 Oct 2024)—is nontrivial and often requires iterative experimentation. Uncertainty quantification using recurrent gated networks can mitigate some real-world effects, outputting confidence levels for predicted force values.

A plausible implication is that combining analytical modeling, sensor calibration, and learning-based adaptation yields superior robustness across a spectrum of operational scenarios—from high-precision surgery and therapy-assistive devices to manufacturing and autonomous agricultural tasks.

7. Impact and Future Directions

End-effector force estimators are increasingly central to modern robotics, enabling physically interactive autonomy, safety-critical haptics, task generalization, and even real-time self-supervision in unstructured environments. Emerging multi-modal sensing and learning frameworks (e.g., MAGPIE’s 8-axis Hall effect architecture with uncertainty-aware gated networks (Tanaka et al., 23 Oct 2024)) point toward highly adaptive, robust solutions for limbed robots and hybrid manipulation/grasping scenarios.

Research trends include task-driven estimator fine-tuning, transfer learning for cross-platform adaptation, continuous bias/drift filtering, and the fusion of proprioceptive, tactile, and exteroceptive data. Persistent limitations—dynamic hysteresis, high-velocity operation, ambiguous contact geometry, and environment-dependent stiffness—remain open research fronts, with ongoing efforts to enhance estimator generality, precision, and responsiveness in real-world deployments.